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Among the myriad of constants that appear in mathematics, p, e, and i are the most familiar. Following closely behind is g, or gamma, a constant that arises in many mathematical areas yet maintains a profound sense of mystery.
In a tantalizing blend of history and mathematics, Julian Havil takes the reader on a journey through logarithms and the harmonic series, the two defining elements of gamma, toward the first account of gamma's place in mathematics.
Introduced by the Swiss mathematician Leonhard Euler (1707-1783), who figures prominently in this book, gamma is defined as the limit of the sum of 1 + 1/2 + 1/3 + . . . up to 1/n, minus the natural logarithm of n--the numerical value being 0.5772156. . .. But unlike its more celebrated colleagues p and e, the exact nature of gamma remains a mystery--we don't even know if gamma can be expressed as a fraction.
Among the numerous topics that arise during this historical odyssey into fundamental mathematical ideas are the Prime Number Theorem and the most important open problem in mathematics today--the Riemann Hypothesis (though no proof of either is offered!).
Sure to be popular with not only students and instructors but all math aficionados, Gamma takes us through countries, centuries, lives, and works, unfolding along the way the stories of some remarkable mathematics from some remarkable mathematicians.
- Sales Rank: #786106 in eBooks
- Published on: 2010-01-04
- Released on: 2010-01-04
- Format: Kindle eBook
Review
"[A] wonderful book. . . . Havil's emphasis on historical context and his conversational style make this a pleasure to read. . . . Gamma is a gold mine of irresistible mathematical nuggets. Anyone with a serious interest in maths will find it richly rewarding."--Ben Longstaff, New Scientist
"This book is a joy from start to finish."--Gerry Leversha, Mathematical Gazette
"[Gamma] is not a book about mathematics, but a book of mathematics. . . . [It] is something like a picaresque novel; the hero, Euler's constant g, serves as the unifying motif through a wide range of mathematical adventures."--Dan Segal, Notices of the American Mathematical Society
"The book is enjoyable for many reasons. Here are just two. First, the explanations are not only complete, but they have the right amount of generality. . . . Second, the pleasure Havil has in contemplating this material is infectious."--Jeremy Gray, MAA Online
"It is only fitting that someone should write a book about gamma, or Euler's constant. Havil takes on this task and does an excellent job."--Choice
"This book is accessible to a wide range of readers, and should particularly appeal to those who feel a love for mathematics and are dissuaded by the dryness and formality of text-books, but are also not satisfied by the less rigorous approach of most popular books. Mathematics is presented throughout as something connected to reality. . . . Many readers will find in this book exactly what they have been missing."--Mohammad Akbar, Plus Magazine, Millennium Mathematics Project, University of Cambridge
"This book is written in an informal, engaging, and often amusing style. The author takes pains to make the mathematics clear. He writes about the mathematical geniuses of the past with reverence and awe. It is especially nice that the mathematical topics are discussed within a historical context."--Ward R. Stewart, Mathematics Teacher
From the Author
Gamma was my first book, written from the inspiration provided by the books by Maor and Nahin on e and i respectively.� It was a considerable suprise to me that my email to PUP suggesting the idea of a similar book on the constant gamma was replied to within days, positively and from Woodstock in England.� It took three years to write.� Now in German and Japanese translation, its paperbck form is part of Princeton Science Library.� My apologies for the typos; it seems that such are inevitable (at least for me) when writing such technical popular books!
From the Inside Flap
"I like this book very much. So much, in fact, that I found myself muttering 'neat stuff!' all the way through. While it is about an important topic, there isn't a single competitor. This amazing oversight by past authors is presumably the result of the topic requiring an author with a pretty sophisticated mathematical personality. Havil clearly has that. His skillful weaving of mathematics and history makes the book a 'fun' read. Many instructors will surely find the book attractive."--Paul J. Nahin, author of Duelling Idiots and Other Probability Puzzlers and An Imaginary Tale
"This is an excellent book, mathematically as well as historically. It represents a significant contribution to the literature on mathematics and its history at the upper undergraduate and graduate levels. Julian Havil injects genuine excitement into the topic."--Eli Maor, author of e: The Story of a Number
Most helpful customer reviews
53 of 54 people found the following review helpful.
Gamma finally joins the ranks of e, pi, i
By Kindle Customer
After reading Paul Nahin's lovely book on i, "An Imaginary Tale" (also published by Princeton University Press), I could not believe it when the same publisher produced a book on gamma. Gamma seems to always have been one of the neglected constants in mathematics (by the general public). e, pi, and i seem to capture the imagination more, my guess is because the mathematics required to understand them are more elementary (I use the word "elementary" completely tongue in cheek), and you can quickly see the dazzling results they are associated with.
Gamma is different. While you can understand the theory presented in Julian Havil's book if you stayed awake during second semester calculus, you definitely have to work at it. The requisite analytic number theory presented may turn away the average reader if they are not prepared to make the commitment to stay on the roller coaster for the full ride.
You will be rewarded if you can break through the initial 2 or 3 chapters introducing us to the logarithm and the harmonic series. To be fair, as a previous reviewer has noted, the material on Napier and the logarithm has been done in a more satisfactory manner by Eli Maor in his book on e. But this is only a minor drawback. As long as you are comfortable with the natural logarithm, you can omit Chapter 1 with no loss.
Chapter 4 starts off with the zeta function, arguably the most enticing and mysterious function in all of mathematics, despite approximately 150 years of analysis by the world's best mathematicians. This one function alone could arguably be said to be the genesis of analytic number theory (even though Dirichlet's work on primes in arithmetic progressions has typically been given credit for that role). All the familiar material is presented, including Euler's product formula, the "trivial" divisors of the zeta function, the infinitude of primes, Euler's evaluation of the zeta function for positive even integer powers, etc.
Of course, the gamma function makes its obligatory appearance. After having read Nahin's book on i, I was initiated into the math connecting the gamma and zeta functions. But Nahin of course could not use Euler-Maclaurin summation or the familiar inequality arguments as this would have taken him too far afield. After having read the traditional fare, such as Hardy-Wright, Apostol, Hua, et al., it was nice to see a more conversational approach to the material. I literally felt like I was sitting in Havil's office while he dissected the material for me, on a level I could comprehend.
My last comments on this book are the extras. As expected, Riemann's hypothesis and complex analysis make extended appearances. I appreciated the fact the Havil resisted the temptation to take the Riemann Hypothesis beyond the traditional mathematical lore and float off into the ethereal. This happened with John Derbyshire's otherwise excellent book "Prime Obsession", which devoted a little too much time to the psychoanalysis of Riemann, who after all, only scratched the surface of this problem. Derbyshire's book is highly recommended though for more material on the Prime Number Theorem, and some of its uses to formulate modern permutations of the Riemann Hypothesis.
He presents the usual anecdotes on Riemann and Hardy (who had a major love affair with the Riemann Hypothesis), but these are sidelines only, as they should be. Also, the material on residue integration and analytic continuation in the appendices is enormously helpful to understand the post Riemann attacks on the problem. In addition, well, it's just pretty mathematics.
The introduction by Freeman Dyson is quite impressive. How many books of popular mathematics get endorsements like that from world-class physicists? The praise is well deserved. This book belongs on every math enthusiast's bookshelf!
46 of 47 people found the following review helpful.
Who would have thought!?
By Palle E T Jorgensen
Who would have thought that there can be so much life in a constant? And one with a Greek name! If you have some math interests, I predict that you will get caught up in the thread of events: They are mathematical topics, but are presented like in a novel or a drama. A book that I couldn't put down. The main characters are the harmonic series, the sub-harmonic series, Riemann's Zeta function, its functional equation, its zeros, the Riemann hypothesis(it is worth a million dollars!), the prime number theorem, (..hard stuff! but it somehow seems easy in this book),Bernoulli numbers, Pell's equation, the distribution of prime numbers.... And if you forgot some of your math, you will have it reviewed in the appendices. They are attractive, well written, and to the point.
67 of 72 people found the following review helpful.
This would make an excellent alternative "Calc III"
By John P. Rickert
I agree wholeheartedly with all the positive comments and enthusiasm that other reviewers have shown. This is a remarkable book, and there should be more like it. I am astounded at how much and what range of mathematics there is in a book of this length and level of accessbility. Which raises a very good point: This would be a superb book for "Calc III". It's unfortunate that many students end their study of mathematics slugging through integration by parts, partial fractions, sequences and series, the logarithm as integral, etc., the traditional hodge-podge of topics called Calculus II. And the ones who progress end up going straight into multivariable calculus with its div, grad, curl, and all that. There is never really any reward for all the work in hacking through Calc II. This book, however, would tie so much of it together, it would all suddenly seem so mysteriously connected and beautiful, and the reader (I hope) would want to go on to Complex Analysis. Thank you, Prof. Havil! I hope you find the proof to the Riemann Hypothesis.
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